The federal government’s role as the backer of most U.S. home loans is becoming entrenched as fiscal issues distract Congress and the White House from a housing-finance overhaul that would shift more risk to private capital, according to lawmakers and analysts.

At the core of such an overhaul is the future of Washington-based Fannie Mae and McLean, Virginia-based Freddie Mac, the so-called government-sponsored enterprises, or GSEs, that provide market liquidity by buying home loans and bundling them into securities. As they neared collapse in 2008, the companies were placed into federal conservatorship.